Yes, probably!
Any business with just a single employee is covered under the Occupational Safety & Health Act of 1970 and subequent revisions. If the business is covered by the Act, then EVERYONE in the business is covered, including the owner veterinarians. On the other hand, self-employed people with NO employees are not covered under the Act so although the safety rules really should be followed, OSHA has no authority to enforce them on non-covered businesses.
The Act is a federal law and is applicable to the entire country so OSHA's rules are federal standards and apply to every business in America. There is a provision in the Act that allows states to administer workplace safety programs if they choose, but their rules and enforcement must be at least as stringent as the federal rules. Currently there are 22 states or terrirories of the US that have assumed the responsibility for enforcement of the standards in private businesses within that state. (See the box on the right for a link to those state plans)
In most cases, the rules in these "state plan states" are identical to the federal rules. In some instances, the state OSHA authority has added some rules that must be followed so businesses in those states shuold spend some time researching their state specifics.
There are 22 states that choose to enforce workplace safety rules over private businesses. There are 6 states that have plans to cover only public (government) workers.
https://www.osha.gov/stateplans/